Exercises: Limit of a function as x → a

Calculate the limit of the following rational function by direct substitution:

$\lim_{x \to 3} \frac{3x^2 - 5x + 2}{x + 2}$

Calculate the following limit by resolving the algebraic indeterminate form:

$\lim_{x \to 2} \frac{x^2 - 4}{x^2 - 3x + 2}$

Determine the one-sided limits of the function $g(x) = \frac{2x + 1}{x - 5}$ as $x$ tends to $5$ and state whether the overall limit exists at this point.

Solve the following limit by multiplying by the appropriate conjugate expression:

$\lim_{x \to 3} \frac{\sqrt{x+1} - 2}{x - 3}$

Calculate the limit of the following expression combining transcendental functions:

$\lim_{x \to 0} \frac{e^x + \cos(x)}{\ln(x+e)}$

Given the following piecewise function:

$f(x) = \begin{cases} 3x - 2 & \text{if } x < 1 \\ x^2 & \text{if } x \ge 1 \end{cases}$

Calculate $\lim_{x \to 1} f(x)$.

Observe the graph of the function $f(x)$. Determine the value of $\lim_{x \to 2} f(x)$ and the value of $f(2)$, and briefly explain the difference between both concepts.

Calculate the following limit by factorizing the polynomials using Ruffini's rule or algebraic identities:

$\lim_{x \to -1} \frac{x^3 + 1}{x^2 - 1}$

The growth rate of a bacteria population in a laboratory experiment is given by the expression $C(t) = \frac{t^2 - 16}{t - 4}$ for $t \neq 4$, where $t$ is the time elapsed in hours. What value does the growth rate approach as the time approaches $4$ hours?

Calculate the value of the parameter $k$ so that the limit of the function $h(x)$ exists at $x = -2$:

$h(x) = \begin{cases} x^2 + kx & \text{if } x < -2 \\ \frac{8}{x} - k & \text{if } x > -2 \end{cases}$

Determine if the limit $\lim_{x \to 3} \frac{|x - 3|}{x - 3}$ exists by first calculating its left-hand and right-hand limits.

The average cost (in dollars) of producing $x$ units of a high-precision electronic component is modeled by the function $C(x) = \frac{\sqrt{x^2 + 100} - 10}{x^2}$. What value does the average cost tend to when the production level $x$ is very small (that is, when it tends to $0$)?

Resolve the following limit which presents a double indeterminate form with radicals in the numerator and in the denominator:

$\lim_{x \to 1} \frac{\sqrt{x+3} - 2}{\sqrt{x} - 1}$

Calculate the one-sided limits and the overall limit (if it exists) of the following rational function at the point where its denominator is zero:

$\lim_{x \to 3} \frac{x - 1}{x^2 - 6x + 9}$

Observe the graph of the function $g(x)$, which represents the temperature profile in a chemical reaction. Determine the one-sided limits at $x = -2$ and the overall limit at $x = 3$.

The position of a particle along a rectilinear path as a function of time $t$ (in seconds) is given by the equation $s(t) = 3t^2 - t$ (in feet). The instantaneous velocity at the instant $t = 2$ is calculated using the limit $\lim_{t \to 2} \frac{s(t) - s(2)}{t - 2}$. Calculate the value of this instantaneous velocity.

Find the values of the parameters $a$ and $b$ so that the limits of the function $f(x)$ exist at the break points $x = 0$ and $x = 3$:

$f(x) = \begin{cases} ax^2 + b & \text{if } x < 0 \\ 2x - a & \text{if } 0 < x < 3 \\ \frac{x^2 - 9}{x - 3} & \text{if } x > 3 \end{cases}$

Determine the values of the parameters $m$ and $n$ knowing that the following equality of limits holds:

$\lim_{x \to -1} \frac{x^2 + mx + n}{x + 1} = 4$

In the structural design of a beam, the shear force $E(x)$ at a distance of $x$ feet from the main support is given by the rational function $E(x) = \frac{x^3 - 5x^2 + 7x - 3}{x^3 - 4x^2 + 5x - 2}$. Calculate the limit of the shear force as $x$ approaches $1$ foot, a critical point where a change in cross-section occurs.

Calculate the following limit involving roots of different indices. (Hint: you can use a suitable change of variable, such as $x = t^6$, to simplify the radicals).

$\lim_{x \to 1} \frac{\sqrt{x} - 1}{\sqrt[3]{x} - 1}$

The concentration of an anesthetic in the bloodstream $t$ hours after its intramuscular administration is given by the mathematical model $C(t) = \frac{1 - \sqrt{1 - t^2}}{t^2}$ for the interval $0 < t < 1$. What is the limit of the concentration as the time $t$ approaches $0$?

Determine the value of the parameter $a > 0$ so that the following condition on the limit is met:

$\lim_{x \to 0} \frac{\sqrt{a + x} - \sqrt{a - x}}{x} = \frac{1}{4}$

From the graph of the function $h(x)$, determine the value of the one-sided limits $\lim_{x \to 1^-} (h(x))^2$ and $\lim_{x \to 1^+} (h(x))^2$. Then, justify whether the overall limit $\lim_{x \to 4} \frac{1}{h(x)}$ exists.

Calculate the one-sided limits of the function $f(x) = \frac{|x^2 - 9|}{x - 3}$ at the points $x = -3$ and $x = 3$. From the results obtained, state whether the overall limits exist at these points.