Question

Evaluate the following limits: \(\lim _{x \rightarrow 6} \frac{x}{\sqrt{1+x^{4}}}\)

Short answer

Question: Evaluate the following limit: \(\lim_{x \rightarrow 6} \frac{x}{\sqrt{1+x^4}}\) Answer: \(\frac{6}{\sqrt{1297}}\)

Step-by-step solution

Identify the limit expression

The expression we need to evaluate is: \(\lim_{x \rightarrow 6} \frac{x}{\sqrt{1+x^4}}\)

Substitute the value of x

We can directly substitute the value of x as 6, as it does not lead to an indeterminate form: \(\frac{6}{\sqrt{1+(6)^4}}\)

Compute the expression

Compute the expression by simplifying the denominator: \(\frac{6}{\sqrt{1+(6)^4}} = \frac{6}{\sqrt{1+1296}} = \frac{6}{\sqrt{1297}}\)

Write the final answer

The final answer is the simplified expression, which is the limit of the given function as x approaches 6: \(\lim_{x \rightarrow 6} \frac{x}{\sqrt{1+x^4}} = \frac{6}{\sqrt{1297}}\)

Key concepts

Substitution Method

When dealing with limit problems in calculus, the substitution method can be your best friend, especially for simple cases. The idea is straightforward: you replace the variable "x" in your expression with its limit value, given that doing so doesn't create an indeterminate form such as \(\frac{0}{0}\) or \(\frac{\infty}{\infty}\).

In the exercise provided, we're evaluating \(\lim_{x \rightarrow 6} \frac{x}{\sqrt{1+x^4}}\). Since direct substitution of \(x = 6\) results in a number that is neither zero nor infinity, it's a clear path to the solution:

• Simply replace \(x\) with 6.
• This gives \(\frac{6}{\sqrt{1+6^4}}\).

This method works well here because the calculation leads straight to a numerical result without any issues.

Continuity

Continuity is a crucial concept in calculus when evaluating limits. A function is continuous at a point if it does not break or jump at that point. In mathematical terms, a function \(f(x)\) is continuous at \(x = c\) if \(\lim_{x \to c} f(x) = f(c)\).

With the original problem, since \(f(x) = \frac{x}{\sqrt{1+x^4}}\), and both the numerator \(x\) and the denominator \(\sqrt{1+x^4}\) are continuous functions (polynomials and radicals over their domains), \(f(x)\) is continuous everywhere in its domain, including at \(x = 6\).

• This continuity justifies that substitution method works without causing an undefined situation.

Thus, by the definition of continuity, the value of the limit is simply the value of the function at the point \(x = 6\).

Simplifying Expressions

Once you've substituted and checked the continuity of the function, the next step is to simplify the expression if possible. This can make it easier to handle and understand, especially in complex cases.

In our problem, after substituting \(x = 6\), it is important to simplify \(\sqrt{1+6^4}\). This "1 plus \(x^4\)" inside a square root might look intimidating, but when broken into steps, it becomes clearer:

• Compute \(6^4\), which equals 1296.
• Add 1 to get 1297.
• The expression becomes \(\frac{6}{\sqrt{1297}}\).

This final step is useful because it gives you the precise value of the limit, which is numeric and can be approximated or left in this simplified form for clarity.